Stability Conditions for Resonant Circuits with Time-Variable Parameters

Parametrically-excited resonant circuits of two-loops or two-node-pairs are dealt with in this paper. Investigation is made on the stability and behavior of the circuits near the four resonant points:
2\Omega_1 = \omega
,
2\Omega_2 = \omega
,
\Omega_1 + \Omega_2 = \omega
, and
|\Omega_1 - \Omega_2| = \omega
, where
\Omega_{1}/2 \pi
and
\Omega_{2}/2 \pi
are the two resonant frequencies and
\omega / 2\pi
is the frequency of parametric excitation. In this paper 1) those circuit conditions are obtained by which the parametricallyexcited linear resonant circuit can be reduced to two independent Mathieu equations; 2) the transition curves from stability to instability at the resonant point near
\Omega_1 + \Omega_2 = \omega
are obtained under a special condition; 3) the stability condition is obtained near the resonant point
\Omega_1 + \Omega_2 = \omega
, under a more general condition, 4) an aperiodic (almost periodic) oscillation is obtained near
\Omega_1 + \Omega_2 = \omega
when the circuit is characterized by a nonlinear differential equation; and 5) it is ascertained that instability does not occur at the resonant point
|\Omega_1 - \Omega_2| = \omega
. Mathematical treatments used here consist of two steps: 1) linear transformation, and 2) an averaging method which reduces a nonlinear nonautonomus differential equation to an autonomous differential equation, assuming that the damping (resistance term), the nonlinearity, and the parametric excitation are all small. Some of the results obtained herein can be applied to a parametrically-excited resonant circuit of more than two-loops or two-node-pairs. No investigation is made for a parametrically-excited resonant circuit with input voltage or input current sources.